on the basis property of an trigonometric functions system of the frankl problem with a nonlocal parity condition in the sobolev space w^2l p (0; π)

In the present paper, we write out the eigenvalues and the corresponding eigenfunctions of the modified frankl problem with a nonlocal parity condition of the first kind. we analyze the completeness, the basis property, and the minimality of the eigenfunctions in the space w^2l p (0; π), where w^2l p (0; π) be the set of functions f 2 w2l p (0; π), satisfying of the following conditions: f^(2k-1)(0) = 0; k = 1; 2; ..., l.